Glashow-Salam-Weinberg Electroweak Model¶

Scope of tutorial¶

In this tutorial, we consider the 1st generation of particles, the fermion-vector boson interactions, the fermion-Higgs boson interactions, the Higgs boson self-interactions, and the vector boson interactions. The quantum chromodynamics (QCD) Lagrangian, which describes the interactions between quarks and gluons, is briefly mentioned at the end.

Introduction¶

In 1934, Enrico Fermi proposed the following 4-fermion model

\begin{aligned} \frac{G_{F}}{\sqrt{2}} ({\bar{ψ}}_{e} γ^{μ} ψ_{\bar{ν}}) ({\bar{ψ}}_{p} γ_{μ} ψ_{n}), \end{aligned}

$\begin{align*} \frac{G_F}{\sqrt{2}}(\bar{\psi}_e \gamma^\mu \psi_{\bar{\nu}})(\bar{\psi}_p \gamma_\mu \psi_n), \end{align*}$ to describe

β

$\beta$ -decay processes such as

n \to p + e^{-} + {\bar{ν}}_{e}

$n \rightarrow p + e^- + \bar{\nu}_e$ . The quantity

\bar{ψ} \equiv ψ^{†} γ^{0}

$\bar{\psi} \equiv \psi^\dagger \gamma^0$ is called the Dirac conjugate of the spinor

ψ

$\psi$ , where the operator

^{†}

$^\dagger$ is the Hermitian conjugate, i.e., the transpose and complex conjugate and

γ^{μ}

$\gamma^\mu$ are

4 \times 4

$4 \times 4$ matrices defined by the anti-commutation relations

γ^{μ} γ^{ν} + γ^{ν} γ^{μ} = 2 g^{μ ν}

$\gamma^\mu \gamma^\nu + \gamma^\nu \gamma^\mu = 2 g^{\mu\nu}$ , where

g^{μ ν}

$g^{\mu\nu}$ is the metric tensor of special relativity.

G_{F}

$G_F$ is called the Fermi constant.

Fermi assumed that $\beta$ -decay involved vector currents, $\bar{\psi} \gamma^\mu \psi$ , but in 1956 T. D. Lee and C. N. Yang noted that no experiment had been performed to test the assumption of parity invariance, that is, the invariance of particle interactions with respect to a reflection in a mirror. They suggested that certain experimental puzzles could be explained if parity was not conserved in the weak interactions. Furthermore, Lee and Yang suggested experiments to test this hypothesis. Shortly thereafter, Chien-Shiung Wu performed the experiment (see https://en.wikipedia.org/wiki/Wu_experiment) and confirmed the violation of parity invariance. The following year, Lee and Yang won the Nobel Prize in Physics. A year later, in 1958, Robert Marshak and his student George Sudarshan developed the vector axial-vector theory (V-A) of the weak interactions, which Richard Feynman and Murray Gell-Mann developed further.

We do not know why the weak interactions violate parity conservation, in particular, why they involve both vector $\bar{\psi} \gamma^\mu \psi$ and axial-vector $\bar{\psi} \gamma^\mu \gamma^5 \psi$ currents, where $\gamma^5 \equiv i \gamma^0 \gamma^1 \gamma^2 \gamma^3$ . Now that we have a better understanding of the weak interactions, however, it would be simpler to say that the weak interactions treat left and right-handed fields differently. But, the V-A jargon has stuck.

The Glashow-Salam-Weinberg (GSW) Langrangian (really a Lagrangian density) is built using the fields below for the 1st generation of leptons and quarks:

\begin{aligned} L & = (\begin{matrix} ν_{L} \\ e_{L} \end{matrix}), e_{R}, \\ Q & = (\begin{matrix} u_{L} \\ d_{L} \end{matrix}), u_{R}, d_{R}, \\ where \\ P_{L} & = \frac{1}{2} (1 - γ^{5}), \\ P_{R} & = \frac{1}{2} (1 + γ^{5}) \\ are projection operators that split a \\ fermion field into its left and right components \\ ψ_{L, R} & = P_{L, R} ψ, \end{aligned}

$\begin{align*} L & = \begin{pmatrix} \nu_L \\ e_L \end{pmatrix},\quad e_R, \\ Q & = \begin{pmatrix} u_L \\ d_L \end{pmatrix},\quad u_R, \quad d_R, \\ & \textrm{where } \\ P_L & = \frac{1}{2}(1 - \gamma^5),\\ P_R & = \frac{1}{2}(1 + \gamma^5) \\ & \textrm{are projection operators that split a} \\ & \textrm{fermion field into its left and right components} \\ \psi_{L,\, R} & = P_{L,\, R} \, \psi , \end{align*}$

which implies $\psi = \psi_L + \psi_R$ . Note the absence, in this model, of the field $\nu_R$ .

The GSW model (before addition of the Higgs field) is defined by the Langrangian,

\begin{aligned} L & = \bar{L} i γ^{μ} (\partial_{μ} 1 + i g_{1} \frac{Y_{L}}{2} B_{μ} 1 + i g_{2} \frac{τ}{2} \cdot W_{μ}) L \\ + \bar{Q} i γ^{μ} (\partial_{μ} 1 + i g_{1} \frac{Y_{L}^{Q}}{2} B_{μ} 1 + i g_{2} \frac{τ}{2} \cdot W_{μ}) Q \\ + {\bar{e}}_{R} i γ^{μ} (\partial_{μ} + i g_{1} \frac{Y_{R}}{2} B_{μ}) e_{R} \\ + {\bar{u}}_{R} i γ^{μ} (\partial_{μ} + i g_{1} \frac{Y_{R}^{u}}{2} B_{μ}) u_{R} \\ + {\bar{d}}_{R} i γ^{μ} (\partial_{μ} + i g_{1} \frac{Y_{R}^{d}}{2} B_{μ}) d_{R} \\ - \frac{1}{4} W_{μ ν} \cdot W^{μ ν} \\ - \frac{1}{4} B_{μ ν} B^{μ ν}, \end{aligned}

$\begin{align*} {\cal L} & = \bar{L} i \gamma^\mu \left(\partial_\mu \mathbf{1} + i g_1 \frac{Y_L}{2} B_\mu \mathbf{1} + i g_2 \frac{\boldsymbol{\tau}}{2} \cdot \mathbf{W}_\mu\right) L\\ & + \bar{Q} i \gamma^\mu \left(\partial_\mu \mathbf{1}+ i g_1 \frac{Y_L^Q}{2} B_\mu \mathbf{1} + i g_2 \frac{\boldsymbol{\tau}}{2} \cdot \mathbf{W}_\mu\right) Q\\ & + \bar{e}_R i \gamma^\mu \left( \partial_\mu + i g_1 \frac{Y_R}{2} B_\mu \right) e_R\\ & + \bar{u}_R i \gamma^\mu \left( \partial_\mu + i g_1 \frac{Y^u_R}{2} B_\mu \right) u_R\\ & + \bar{d}_R i \gamma^\mu \left( \partial_\mu + i g_1 \frac{Y^d_R}{2} B_\mu \right) d_R \\ & - \frac{1}{4} \mathbf{W}_{\mu\nu} \cdot \mathbf{W}^{\mu\nu}\\ & - \frac{1}{4} B_{\mu\nu} B^{\mu\nu}, \end{align*}$

where

\begin{aligned} 1 & is a 2 \times 2 unit matrix, \\ τ & are the Pauli matrices, \\ the Y & are called hypercharges, \\ while & the tensors W_{μ ν} and B_{μ ν} are given by \\ W_{μ ν} & = \partial_{μ} W_{ν} - \partial_{ν} W_{μ} - g_{2} W_{μ} \times W_{ν}, \\ (W_{μ ν})_{i} & = (\partial_{μ} W_{ν})_{i} - (\partial_{ν} W_{μ})_{i} - g_{2} ϵ_{i j k} W_{μ}^{j} W_{ν}^{k}, \\ B_{μ ν} & = \partial_{μ} B_{ν} - \partial_{ν} B_{μ} and \\ W_{μ} & = (W_{μ}^{1}, W_{μ}^{2}, W_{μ}^{3}) . \end{aligned}

$\begin{align*} \mathbf{1} & \textrm{ is a $2 \times 2$ unit matrix}, \\ \boldsymbol{\tau} & \textrm{ are the Pauli matrices},\\ \textrm{the } Y & \textrm{ are called hypercharges},\\ \textrm{while }& \textrm{the tensors $W_{\mu\nu}$ and $B_{\mu\nu}$ are given by}\\ \mathbf{W}_{\mu\nu} & = \partial_\mu \mathbf{W}_\nu- \partial_\nu \mathbf{W}_\mu - g_2 \, \mathbf{W}_\mu \times \mathbf{W}_\nu,\\ (\mathbf{W}_{\mu\nu})_i & = (\partial_\mu \mathbf{W}_\nu)_i - (\partial_\nu \mathbf{W}_\mu)_i - g_2 \, \epsilon_{ijk} W^j_\mu W^k_\nu, \\ B_{\mu\nu} & = \partial_\mu B_\nu - \partial_\nu B_\mu \textrm{ and }\\ \mathbf{W}_\mu & = (W^1_\mu, \, W^2_\mu, \, W^3_\mu). \end{align*}$

The quantities enclosed in the first and second pair of parentheses in the Lagrangian are $2 \times 2$ matrices that act on the lepton and quark doublet fields $L$ and $Q$ , respectively, while the parentheses in the remaining fermionic terms enclose $1 \times 1$ matrices, that is, scalars, which act on the singlet fields $e_R$ , $u_R$ , and $d_R$ , respectively. However, each field is intrinsically a 4-component Lorentz object: a 4-component spinor in the case of the fermionic fields and a 4-vector in the case of the bosonic fields.

In the GSW Lagrangian, the hypercharges are assigned specific numerical values. Therefore, the Lagrangian has only two free parameters, namely, the couplings $g_1$ and $g_2$ .

$SU(2) \otimes U(1)$ Invariance¶

The GSW Lagrangian is invariant under the group $SU(2) \otimes U(1)$ . The elements $U$ of $SU(2)$ can be represented as the $2 \times 2$ matrices

\begin{aligned} U & = \exp (i (τ / 2) \cdot θ (x)), \\ while \\ U & = \exp (i (Y / 2) α (x)), \end{aligned}

$\begin{align*} U & = \exp(i (\boldsymbol{\tau} \, / \, 2) \cdot \boldsymbol{\theta}(x)), \\\textrm{ while}\\ U & = \exp(i (Y \, / \, 2) \, \alpha(x) ), \end{align*}$

represent the elements of $U(1)$ . The $SU(2)$ elements are defined by three spacetime-dependent functions, $\mathbf{\theta}(x) = (\theta_1(x)$ , $\theta_2(x)$ , $\theta_3(x))$ , each associated with group generators $T_1 = \tau_1 / 2$ , $T_2 = \tau_2 / 2$ , and $T_3 = \tau_3 / 2$ , respectively, while the $U(1)$ elements are defined by a single function $\alpha(x)$ and a single generator $Y \, / \, 2$ .

Invariance under the group $SU(2) \otimes U(1)$ means that the following replacements leave the Lagrangian unchanged,

\begin{aligned} S U (2) \\ L & \to U L, Q \to U Q, \\ e_{R} & \to e_{R}, u_{R} \to u_{R}, d_{R} \to d_{R} \\ \frac{1}{2} τ \cdot W_{μ} & \to U (\frac{1}{2} τ \cdot W_{μ}) U^{- 1} - \frac{1}{i g_{2}} (\partial_{μ} U) U^{- 1}, \\ B_{μ} & \to B_{μ}, \\ U (1) \\ L & \to U L, Q \to U Q \\ e_{R} & \to U e_{R}, u_{R} \to U u_{R}, d_{R} \to U d_{R}, \\ W_{μ} & \to W_{μ}, \\ B_{μ} & \to U B_{μ} U^{- 1} - \frac{1}{i g_{1}} (\partial_{μ} U) U^{- 1} . \end{aligned}

$\begin{align*} \boldsymbol{SU(2)} \quad \quad \\ L & \rightarrow U L, \quad Q \rightarrow U Q, \\ e_R & \rightarrow e_R, \quad u_R \rightarrow u_R, \quad d_R \rightarrow d_R \\ \frac{1}{2} \boldsymbol{\tau} \cdot \mathbf{W}_\mu & \rightarrow U \left( \frac{1}{2} \boldsymbol{\tau} \cdot \mathbf{W}_\mu \right) U^{-1} - \frac{1}{i g_2} (\partial_\mu U) U^{-1}, \\ B_\mu & \rightarrow B_\mu, \\ \boldsymbol{U(1)} \quad \quad \\ L & \rightarrow U L, \quad Q \rightarrow U Q \\ e_R & \rightarrow U e_R, \quad u_R \rightarrow U u_R, \quad d_R \rightarrow U d_R, \\ \mathbf{W}_\mu & \rightarrow \mathbf{W}_\mu, \\ B_\mu & \rightarrow U B_\mu U^{-1} - \frac{1}{i g_1} (\partial_\mu U) U^{-1}. \end{align*}$

It is usually argued that the above local gauge invariance means that the phases of the fields can be set to different values at different events in spacetime and that this makes sense because there is no reason why, for example, Andromedans, ten million years from now, should adopt the same phase convention as Earthlings do now. Personally, I find this argument unconvincing because there are, after all, global symmetries that span the whole of spacetime. For example, the replacements (see for example, https://arxiv.org/pdf/hep-ph/0410370.pdf, p. 9)

\begin{aligned} Q & \to e^{i θ / 3} Q, \\ u_{R} & \to e^{i θ / 3} u_{R}, \\ d_{R} & \to e^{i θ / 3} d_{R}, \\ and \\ L & \to e^{i ϕ} L, \\ e_{R} & \to e^{i ϕ} e_{R}, \end{aligned}

$\begin{align*} Q & \rightarrow e^{i \theta / 3} Q,\\ u_R & \rightarrow e^{i \theta / 3} u_R, \\ d_R & \rightarrow e^{i \theta / 3} d_R, \\ \textrm{and} & \\ L & \rightarrow e^{i \phi } L, \\ e_R & \rightarrow e^{i \phi } e_R, \end{align*}$

which correspond to baryon number and lepton number conservation, respectively, are global symmetries that leave the Lagrangian invariant. These global symmetries were not imposed, but appear for free and, in that sense, are accidental. My question is: why is it fine to have global symmetries where certain quantities, such as $\theta$ and $\phi$ must be the same everywhere and everywhen, that is, Andromedans and Earthlings must use the same values of the phases even though they have never met, and yet it is not fine to have the same gauge field phases everywhere? There is no good reason that I can discern. I think it is better simply to accept the discovery that local guage invariance yields theories, which for reasons not yet clear, are spectacularly successful.

Hypercharges¶

The hypercharge $Y$ , electric charge $Q$ (in units of the proton charge and not to be confused with the quark doublet), and the third component of the isospin $I_3 = \pm 1/2$ are related by the Gell-Mann Nishijima relation $Y = 2(Q - I_3)$ . For the left-handed neutrino field, $Q = 0$ and $I_3 = +1/2$ , therefore, the hypercharge is $Y_L = -1$ . For the left-handed electron field, $Q = -1$ and $I_3 = -1/2$ and again $Y_L = -1$ . On the other hand, since the right-handed electron field is an $SU(2)$ singlet and, therefore, transforms trivially, that is, $e_R \rightarrow e_R$ , $I_3 = 0$ and therefore $Y_R = -2$ . We conclude that the hypercharge is a property of the doublet or singlet.

In the quark model, up quarks have electric charge $Q = +2/3$ (in units of the proton charge), while down quarks have electric charge $Q = -1/3$ . Therefore, in order to satisfy the Gell-Mann Nishijima relation we must make the assignments $Y^Q_L = 1/3$ , $Y^u_R = 4/3$ , and $Y^d_R = -2/3$ .

Bosons¶

If we define $X_\mu = (\boldsymbol{\tau}/2) \cdot \mathbf{W}_\mu$ and $X_{\mu\nu} = (\boldsymbol{\tau}/2) \cdot\mathbf{W}_{\mu\nu}$ , and noting that $\tau^a \tau^b = i \epsilon_{abc} \tau^c + \delta_{ab}\mathbf{1}$ and, therefore, $\textrm{Tr}(\tau^a \tau^b) = 2 \delta_{ab}$ , we can write

\begin{aligned} X_{μ ν} & = \partial_{μ} X_{ν} - \partial_{ν} X_{μ} + i g_{2} [X_{μ}, X_{ν}], \end{aligned}

$\begin{align*} X_{\mu\nu} & = \partial_\mu X_\nu - \partial_\nu X_\mu + i g_2 [X_\mu, X_\nu] , \end{align*}$ and, therefore,

\begin{aligned} W_{μ ν} \cdot W^{μ ν} & = \frac{1}{2} Tr (X_{μ ν} X^{μ ν}) . \end{aligned}

$\begin{align*} \mathbf{W}_{\mu\nu} \cdot \mathbf{W}^{\mu\nu} & = \frac{1}{2}\textrm{Tr}(X_{\mu\nu} X^{\mu\nu}) . \end{align*}$

In the low energy limit ( $q^2 << M_W$ , where the W boson mass $M_W \sim 80\,$ GeV), the GSW model reproduces the Fermi model for $\beta$ -decay provided that we set

\frac{G_{F}}{\sqrt{2}} = \frac{g_{2}^{2}}{8 M_{W}^{2}} .

$\frac{G_F}{\sqrt{2}} = \frac{g_2^2}{8 M_W^2}.$

import sympy as sm
import sympy.functions.special.tensor_functions as t
from sympy.matrices import Matrix
from sympy.physics.matrices import mgamma, msigma
from sympy import I, trigsimp, radsimp

# enable pretty printing of equations
sm.init_printing()

levi = t.LeviCivita
kron = t.KroneckerDelta

Define fields¶

Alas, there are differing conventions for defining the fields. In the textbook by Mark Thomson, $W^\pm_\mu = \frac{1}{\sqrt{2}} (W^1_\mu \mp W^2_\mu)$ , while in the textbook by Gordon Kane (Modern Elementary Particle Physics, 2nd Edition) these fields are defined by $W^\pm_\mu = -\frac{1}{\sqrt{2}} (W^1_\mu \pm W^2_\mu)$ . And in the book by Donnelly, Formaggio, Holstein, Milner, and Surrow (Foundations of Nuclear and Particle Physics), the convention is $W^\pm_\mu = \frac{1}{\sqrt{2}} (W^1_\mu \pm W^2_\mu)$ . Here, we'll follow the convention in Thomson's book.

gamma  = sm.Symbol('\gamma^{\mu}', commutative=False)
nubarL = sm.Symbol('\overline{\\nu}_L', commutative=False)
nuL    = sm.Symbol('\\nu_L', commutative=False)
ebarL  = sm.Symbol('\overline{e}_L', commutative=False)
eL     = sm.Symbol('e_L', commutative=False)
ebarR  = sm.Symbol('\overline{e}_R', commutative=False)
eR     = sm.Symbol('e_R', commutative=False)

ubarL  = sm.Symbol('\overline{u}_L', commutative=False)
uL     = sm.Symbol('u_L', commutative=False)
ubarR  = sm.Symbol('\overline{u}_R', commutative=False)
uR     = sm.Symbol('u_R', commutative=False)

dbarL  = sm.Symbol('\overline{d}_L', commutative=False)
dL     = sm.Symbol('d_L', commutative=False)
dbarR  = sm.Symbol('\overline{d}_R', commutative=False)
dR     = sm.Symbol('d_R', commutative=False)

sbarL  = sm.Symbol('\overline{s}_L', commutative=False)
sL     = sm.Symbol('s_L', commutative=False)
sbarR  = sm.Symbol('\overline{s}_R', commutative=False)
sR     = sm.Symbol('s_R', commutative=False)

cbarL  = sm.Symbol('\overline{c}_L', commutative=False)
cL     = sm.Symbol('c_L', commutative=False)
cbarR  = sm.Symbol('\overline{c}_R', commutative=False)
cR     = sm.Symbol('c_R', commutative=False)

B, W1, W2, W3         = sm.symbols("B_\mu, W^1_\mu, W^2_\mu, W^3_\mu")
Bnu, W1nu, W2nu, W3nu = sm.symbols("B_\\nu, W^1_\\nu, W^2_\\nu, W^3_\\nu")
Wplus, Wminus         = sm.symbols("W^+_\mu, W^-_\mu")
Wplusnu,Wminusnu,W3nu = sm.symbols("W^+_\\nu, W^-_\\nu, W^3_\\nu")

Z, A     = sm.symbols('Z_\mu, A_\mu')
Znu, Anu = sm.symbols('Z_\\nu, A_\\nu')

nuL, eL, eR, uL, uR, dL, dR, B, W1, W2, W3

Wplus, Wminus, Z, A

Define couplings, hypercharges, and weak mixing angle $\theta_W$ ¶

g1, g2, thetaW     = sm.symbols('g_1 g_2 \\theta_W')
YQL, YdL, YuR, YdR = sm.symbols('Y^Q_L Y^d_L Y^u_R Y^d_R')
YL, YR, YphiL      = sm.symbols('Y_L Y_R Y_L^\phi')
g1, g2, YL, YR, YQL, YuR, YdR, YphiL, thetaW

Define $SU(2)$ lepton and quark doublets $L$ and $Q$ ¶

L = Matrix([nuL, eL])
Lbar = Matrix([nubarL, ebarL]).T

Q = Matrix([uL, dL])
Qbar = Matrix([ubarL, dbarL]).T

Lbar, L, Qbar, Q

Define $\tau \cdot W$ matrix¶

tau1 = msigma(1)
tau2 = msigma(2)
tau3 = msigma(3)
w    = tau1*W1 + tau2*W2 + tau3*W3
w

The fields $W^1_\mu \mp W^2_\mu$ have the form of charged fields, which we write as $(W^1_\mu \mp W^2_\mu) \, / \, \sqrt{2} \rightarrow W^\pm_\mu$ , and are identified as the fields of the charged weak currents.

w = w.subs({W1 - I*W2: sm.sqrt(2)*Wplus,
            W1 + I*W2: sm.sqrt(2)*Wminus})
w

Define matrix $b$ ¶

b = B*tau1**2 
b

Compute matrices¶

\begin{aligned} M_{L} & = g_{1} \frac{Y_{L}}{2} B_{μ} + g_{2} \frac{τ}{2} \cdot W_{μ} \end{aligned}

$\begin{align} M_L & = g_1 \frac{Y_L}{2} B_\mu + g_2 \frac{\boldsymbol{\tau}}{2} \cdot \mathbf{W}_\mu \end{align}$

and

\begin{aligned} M_{Q} & = g_{1} \frac{Y_{L}^{Q}}{2} B_{μ} + g_{2} \frac{τ}{2} \cdot W_{μ} \end{aligned}

$\begin{align} M_Q & = g_1 \frac{Y^Q_L}{2} B_\mu + g_2 \frac{\boldsymbol{\tau}}{2} \cdot \mathbf{W}_\mu \end{align}$

ML = (g1/2)*YL*b + (g2/2)*w

MQ = (g1/2)*YQL*b + (g2/2)*w

ML, MQ

Expand¶

$- \bar{L} \gamma^\mu M_L L - \bar{Q} \gamma^\mu M_Q Q$

Note: we had to switch the order of gamma and Lbar for the algebra to look correct.

a = sm.expand(-gamma*Lbar*ML*L)[0] + sm.expand(-gamma*Qbar*MQ*Q)[0]
a

Add the singlet terms¶

\begin{aligned} - \frac{g_{1}}{2} Y_{R} B_{μ} {\bar{e}}_{R} γ^{μ} e_{R} - \frac{g_{1}}{2} Y_{R}^{u} B_{μ} {\bar{u}}_{R} γ^{μ} u_{R} - \frac{g_{1}}{2} Y_{R}^{d} B_{μ} {\bar{d}}_{R} γ^{μ} d_{R} \end{aligned}

$\begin{align*} - \frac{g_1}{2} Y_{R} B_{\mu} \bar{e}_R \gamma^\mu e_R - \frac{g_1}{2} Y^u_{R} B_{\mu} \bar{u}_R \gamma^\mu u_R - \frac{g_1}{2} Y^d_{R} B_{\mu} \bar{d}_R \gamma^\mu d_R \end{align*}$

l = a
l = l - (g1/2)*YR*B*ebarR*gamma*eR 
l = l - (g1/2)*YuR*ubarR*gamma*B*uR
l = l - (g1/2)*YdR*dbarR*gamma*B*dR
l

The unification of the electromagnetic and weak nuclear forces¶

The key idea in the GSW model, as in the original 1961 model by Glashow, is the unification of the electromagnetic and weak interations, which is expressed as follows

\begin{aligned} (\begin{matrix} B_{μ} \\ W_{μ}^{3} \end{matrix}) & = (\begin{matrix} \cos θ_{W} & - \sin θ_{W} \\ \sin θ_{W} & \cos θ_{W} \end{matrix}) (\begin{matrix} A_{μ} \\ Z_{μ} \end{matrix}) . \end{aligned}

$\begin{align} \begin{pmatrix} B_\mu \\ W^3_\mu \end{pmatrix} & = \begin{pmatrix} \cos\theta_W & -\sin\theta_W \\ \sin\theta_W & \cos\theta_W \end{pmatrix} \begin{pmatrix} A_\mu \\ Z_\mu \end{pmatrix} . \end{align}$

What the unification means concretely is that above a certain critical temperature, the symmetry between the two forces (called the electroweak symmetry) is restored and the two forces merge into a single electroweak force. Below the critical temperature, the electroweak symmetry is said to be spontaneously broken by the change in shape of the potential energy of the Higgs field. While the ensemble of infinitely many vacuum state solutions of the theory respects the electroweak symmetry, the particular vacuum state into which the universe settles does not.

We should be thankful for broken symmetry because without it the universe would be far less diverse and we would not exist.

AZv = Matrix([A, Z])
BWv = Matrix([B, W3])
rot = Matrix([[sm.cos(thetaW),  sm.sin(thetaW)],
             [-sm.sin(thetaW),  sm.cos(thetaW)]])
irot= rot**-1
irot.simplify()
BW  = irot * AZv
AZ  = rot  * BWv
AZ, BW

Collect terms for $A_\mu$ and $Z_\mu$ ¶

l = l.subs(B,  BW[0])
l = l.subs(W3, BW[1])
l = sm.expand(l)
l = sm.collect(l, A)
l = sm.collect(l, Z)
l

In the above expression there is an interaction between the electromagnetic field $A_\mu$ and the neutrino current $\bar{\nu}_L \gamma^\mu \nu_L$ . However, we have no evidence that such an interaction exists. Therefore, we need to get rid of it. This can be done by setting $-Y_L \, g_1\cos\theta_W$ $=$ $g_2\sin\theta_W$ .

l = l.subs(-YL*g1*sm.cos(thetaW), g2*sm.sin(thetaW))
l

So far, so good. Now we wish to recover the known form of the electromagnetic interaction, which for electrons is $q \, A_\mu (\bar{e}_L \gamma^\mu e_L + \bar{e}_R \gamma^\mu e_R )$ . We can do this if we impose the conditions

\begin{aligned} - (Y_{R} / 2) g_{1} \cos θ_{W} & = g_{2} \sin θ_{W} = q, \end{aligned}

$\begin{align*} -(Y_R / 2) g_1 \cos\theta_W & = g_2 \sin\theta_W = q, \end{align*}$

where $q$ is the electric charge, or equivalently,

\begin{aligned} \cos θ_{W} & = - \frac{q}{g_{1} (Y_{R} / 2)} and \\ \sin θ_{W} & = \frac{q}{g_{2}} . \end{aligned}

$\begin{align*} \cos\theta_W & = -\frac{q}{g_1 (Y_R/2)} \, \textrm{ and }\\ \sin\theta_W & = \frac{q}{g_2}. \end{align*}$

The above expressions imply

\begin{aligned} q & = \frac{g_{1} g_{2} (Y_{R} / 2)}{\sqrt{g_{1}^{2} (Y_{R} / 2)^{2} + g_{2}^{2}}}, \\ \tan θ_{W} & = - (Y_{R} / 2) \frac{g_{1}}{g_{2}} \end{aligned}

$\begin{align*} q & = \frac{g_1 g_2 (Y_R/2)}{\sqrt{g_1^2 (Y_R/2)^2 + g_2^2}}, \\ \tan\theta_W & = -(Y_R/2)\frac{g_1}{g_2} \end{align*}$

l = l.subs({-YR*g1*sm.cos(thetaW): 2*g2*sm.sin(thetaW) })
l

Simplify expressions¶

We extract the coefficients of the bosonic fields, that is, the currents, and manipulate them into the forms in which they are typically expressed. One of the key manipulations is substituting the appropriate expressions for the hypercharges.

Hypercharge assignments¶

Assume the validity of the relation $Y = 2(Q - I_3)$ , where $Q$ and $I_3$ , respectively, are the electric charge in units of the proton charge and the 3rd component of the isospin. Setting $Q = 0$ for the upper component of the lepton doublet, $L$ , which is occupied by the neutrino field, and noting that $I_3 = +1/2$ for that component, we find $Y_L = -1$ . Repeating this for the lower component, $Q = -1$ and $I_3 = -1/2$ , we again obtain $Y_L = -1$ . And similarly for $Y_R$ with $Q = -1$ and $I_3 = 0$ , we find $Y_R = -2$ . Also, $Y^Q_L = 2 (Q^f - I^f_3) = 1/3$ , $Y^u_R = 2 Q^u = 4/3$ , where $f$ is the fermion flavor, and $Y^d_R = 2 Q^d = -2/3$ .

q = sm.symbols('q')
Qf, Qu, Qd, Qe, Qnu = sm.symbols('Q^f, Q^u, Q^d, Q^e, Q^\\nu')
If3, Id3, Iu3, Ie3, Inu3 = sm.symbols('I^f_3, I^d_3, I^u_3, I^e_3, I^\\nu_3')
q, Qf, Qu, Qd, Qe, Qnu, If3, Id3, Iu3, Ie3, Inu3

Get coefficient of $W^+_\mu$ ¶

Wpluscoeff  = l.coeff(Wplus)
Wpluscoeff  = Wpluscoeff.subs(g2, q/sm.sin(thetaW)).factor(q).simplify()
Wpluscoeff

Get coefficient of $W^-_\mu$ ¶

Wminuscoeff = l.coeff(Wminus)
Wminuscoeff = Wminuscoeff.subs(g2, q/sm.sin(thetaW)).factor(q).simplify()
Wminuscoeff

Get coefficient of $A_\mu$ ¶

N, E, U, D = sm.symbols('N, E, U, D')

Acoeff = l.coeff(A).expand()

Acoeff = Acoeff.subs({ebarL*gamma*eL: E,
                      ubarL*gamma*uL: U, 
                      dbarL*gamma*dL: D})
Acoeff = Acoeff.collect(E).collect(U).collect(D)

Acoeff = Acoeff.subs({E:ebarL*gamma*eL, 
                      U: ubarL*gamma*uL, 
                      D: dbarL*gamma*dL})

Acoeff = Acoeff.subs(g1, g2*sm.sin(thetaW)/sm.cos(thetaW))

Acoeff = Acoeff.subs({g2*sm.sin(thetaW): q})

Acoeff

Implement hypercharge assignments for electromagnetic current.

Acoeff = Acoeff.subs(YuR, 2*Qu)
Acoeff = Acoeff.subs(YdR, 2*Qd)
Acoeff = Acoeff.subs(YQL, 2*(Qf - If3))
Acoeff

# up quark current
old = Acoeff.coeff(ubarL*gamma*uL)
new = old.subs({2*If3: 1, Qf: Qu})
Acoeff = Acoeff.subs({old: new})

# down quark current
old = Acoeff.coeff(dbarL*gamma*dL)
new = old.subs({2*If3:-1, Qf: Qd})
Acoeff = Acoeff.subs({old: new}).simplify()

Acoeff

From the above that the electromagnetic current $J^\mu_\textrm{em}$ can be written as

\begin{aligned} J_{em}^{μ} & = - q \sum_{f = u, d, e} Q^{f} {\bar{ψ}}^{f} γ^{μ} ψ^{f}, \end{aligned}

$\begin{align*} J^\mu_\textrm{em} & = - q \sum_{f = u, d, e} Q^f \bar{\psi}^f \gamma^\mu \psi^f, \end{align*}$ where

ψ = ψ_{L} + ψ_{R}

$\psi = \psi_L + \psi_R$ , noting that

P_{L, R} γ^{μ} P_{L, R} = 0

$P_{L,R} \gamma^\mu P_{L, R} = 0$ .

Get coefficient of $Z_\mu$ ¶

Try some algebraic manipulations to simplify expression. Unfortunately, collect does not work on non-commutative symbols. So try the following hack.

Zcoeff = l.coeff(Z)

Zcoeff = Zcoeff.subs({nubarL*gamma*nuL: N, 
                      ebarL*gamma*eL: E,
                      ubarL*gamma*uL: U, 
                      dbarL*gamma*dL: D})
Zcoeff = Zcoeff.collect(E).collect(N).collect(U).collect(D)

Zcoeff = Zcoeff.subs({N:nubarL*gamma*nuL, 
                      E:ebarL*gamma*eL, 
                      U: ubarL*gamma*uL, 
                      D: dbarL*gamma*dL})
Zcoeff = Zcoeff.subs(g1, g2*sm.sin(thetaW)/sm.cos(thetaW))

Zcoeff

Notice that the coefficients of $g_2 \cos(\theta_W)$ is $-I_3$ . Therefore, let's make those substitutions in the above.

old = Zcoeff.coeff(ubarL*gamma*uL)
new = old.subs({g2*sm.cos(thetaW)/2: g2*Iu3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(dbarL*gamma*dL)
new = old.subs({g2*sm.cos(thetaW)/2: -g2*Id3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(nubarL*gamma*nuL)
new = old.subs({g2*sm.cos(thetaW)/2: g2*Inu3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarL*gamma*eL)
new = old.subs({g2*sm.cos(thetaW)/2: -g2*Ie3*sm.cos(thetaW)}).factor(g2)
Zcoeff = Zcoeff.subs({old: new})

Zcoeff

Implement hypercharge assignments for weak neutral current. Note the $I_3$ value for singlet fields is always zero.

Tip: To disable a cell use esc r; use esc y to reactivate it and esc m to go to markdown mode.

old = Zcoeff.coeff(dbarR*gamma*dR)
new = old.subs({YdR: 2*Qd}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ubarR*gamma*uR)
new = old.subs({YuR: 2*Qu}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarR*gamma*eR)
new = old.subs({YR: 2*Qe}).cancel().factor(g2)
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(nubarL*gamma*nuL)
new = old.subs({YL: 2*(Qnu - Inu3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(dbarL*gamma*dL)
new = old.subs({YQL: 2*(Qd - Id3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ebarL*gamma*eL)
new = old.subs({YL: 2*(Qe - Ie3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

old = Zcoeff.coeff(ubarL*gamma*uL)
new = old.subs({YQL: 2*(Qu - Iu3)}).simplify()
Zcoeff = Zcoeff.subs({old: new})

Zcoeff

The weak neutral current $J^\mu_\textrm{nc}$ can be written as

\begin{aligned} J_{nc}^{μ} & = \frac{2 q}{\sin (2 θ_{W})} (\sum_{f = d, e, u} Q^{f} \sin^{2} θ_{W} {\bar{ψ}}_{R}^{f} γ^{μ} ψ_{R}^{f} + \sum_{f = ν, d, e, u} (- I_{3}^{f} + Q^{f} \sin^{2} θ_{W}) {\bar{ψ}}_{L}^{f} γ^{μ} ψ_{L}^{f}) . \end{aligned}

$\begin{align*} J^\mu_\textrm{nc} & = \frac{2q}{\sin(2\theta_W)} \left( \sum_{f = d, e, u} Q^f \sin^2\theta_W \bar{\psi}^f_R \gamma^\mu \psi^f_R + \sum_{f = \nu, d, e, u} (-I^f_3 + Q^f \sin^2\theta_W) \, \bar{\psi}^f_L \gamma^\mu \psi^f_L \right). \end{align*}$

Final form of GSW lagrangian for 1st generation fermions¶

Electromagnetic interactions¶

Acoeff*A

Weak charged current interactions¶

Wpluscoeff*Wplus + Wminuscoeff*Wminus

Weak neutral current interactions¶

Zcoeff = Zcoeff.factor(g2).subs({g2: q / sm.sin(thetaW)})
Zcoeff*Z

The GSW model incorporates quantum electrodynamics (QED), predicts weak charged and neutral current interactions and incorporates the observation that neither the electromagnetic nor the weak neutral currrent interactions change flavor, while the weak charged current interactions always do. And, of course, the model incorporates the parity violation of the weak interactions.

The Higgs Sector¶

The Higgs Lagrangian is given by

\begin{aligned} L_{Higgs} & = (D^{μ} ϕ)^{†} (D_{μ} ϕ) - V (ϕ^{†} ϕ), \end{aligned}

$\begin{align} {\cal L}_\textrm{Higgs} &= (D^\mu \phi)^\dagger (D_\mu \phi) - V(\phi^\dagger \phi) , \end{align}$ where

\begin{aligned} D_{μ} & = \partial_{μ} + i g_{1} \frac{Y_{L}^{ϕ}}{2} B_{μ} + i g_{2} \frac{τ}{2} \cdot W_{μ}, \\ V (ϕ^{†} ϕ) & = μ^{2} ϕ^{†} ϕ + λ (ϕ^{†} ϕ)^{2}, and the 4-component Higgs field is given by \\ ϕ & = (\begin{matrix} ϕ_{1} + i ϕ_{2} \\ ϕ_{3} + i ϕ_{4} \end{matrix}) . \end{aligned}

$\begin{align} D_\mu &= \partial_\mu + i g_1 \frac{Y^\phi_L}{2} B_\mu + i g_2 \frac{\boldsymbol\tau}{2} \cdot \mathbf{W}_\mu,\\ V(\phi^\dagger \phi) &= \mu^2 \phi^\dagger \phi + \lambda (\phi^\dagger \phi)^2, \quad\textrm{and the 4-component Higgs field is given by}\\ \phi & = \begin{pmatrix} \phi_1 + i \phi_2 \\ \phi_3 + i \phi_4 \end{pmatrix} . \end{align}$

Qualitative description of the Higgs mechanism¶

The stability of the system requires $\lambda > 0$ . In the very early universe, we presume that $\mu^2 > 0$ , in which case $\langle 0| \phi | 0 \rangle = 0$ , that is, the value of the Higgs field in the vacuum is zero. But, at some critical temperature, $T_\textrm{crit}$ , there is a phase transition in which $\mu^2$ changes sign thereby causing the Higgs potential to develop infinitely many degenerate vacua in each of which the Higgs field is non-zero. Above $T_\textrm{crit}$ , there is a unique vacuum state and this state respects the symmetry of the Lagrangian. Below $T_\textrm{crit}$ , the ensemble of vacua respects the symmetry, but, the universe cools into one of the infinitely many vacua, which on its own breaks the symmetry. This is an example of spontaneous symmetry breaking, here electroweak symmetry breaking (EWSB). Spontaneous symmetry breaking is entirely analogous to the breaking of the $O(3)$ symmetry in an (infinite) ferromagnet as it cools below its critical temperature and develops a magnetization in one of infinitely many possible directions, that is, "vacuum" states of the ferromagnet.

The vacuum state of the universe is one in which the Higgs field behaves like a superconductor that attenuates the propagation of the $W_\mu$ and $Z_\mu$ fields, thereby rendering these fields massive, but permits the free propagation of the $A_\mu$ . (The theory is made to do this!) Moreover, the interactions between the Higgs field and the fermions cause a rapid flipping between the left-handed and right-handed fermion fields, which builds up energy within them that we interpret as their mass. The results from the LHC experiment are, so far, consistent with this picture.

Since there is no right-handed neutrino the flipping mechanism cannot occur and therefore the neutrino remains massless in the GSW model. However, neutrinos are now known to have (very small) masses, but, how their masses arise remains an open question.

Quantum numbers¶

We need to specify the value of the hypercharge $Y^\phi_L$ associated with the doublet $\phi$ . As above, the hypercharge is chosen so that the Gell-Mann Nishijima relation $Y = 2(Q - I_3)$ holds, where the third component of the weak isospin $I_3 = +1/2$ for the upper component of a doublet and $I_3 = -1/2$ for the lower component. In the GSW model, it is assumed that the vacuum expectation of $\phi$ is

⟨ 0 | ϕ | 0 ⟩ = \frac{1}{\sqrt{2}} (\begin{matrix} 0 \\ v \end{matrix}) .

$\langle 0|\phi|0 \rangle = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v \end{pmatrix}.$ Since the vacuum has a net electric charge of zero, we must set

Q = 0

$Q = 0$ for the lower component

ϕ_{3} + i ϕ_{4}

$\phi_3 + i \phi_4$ . From

Y_{L}^{ϕ} = 2 (Q - I_{3})

$Y^\phi_L = 2 (Q - I_3)$ it follows that

Y_{L}^{ϕ} = + 1

$Y^\phi_L = +1$ . This also shows that the upper component,

ϕ_{1} + i ϕ_{2}

$\phi_1 + i \phi_2$ must be assigned

Q = + 1

$Q = +1$ , which of course is why it is assumed to be zero in the vacuum otherwise the entire universe would have a huge net positive charge!

Higgs field at low energies¶

At energies not too far from the vacuum state, we assume that the Higgs field can be approximated as follows

ϕ \approx H = \frac{1}{\sqrt{2}} (\begin{matrix} 0 \\ v + h \end{matrix}),

$\phi \approx H = \frac{1}{\sqrt{2}}\begin{pmatrix} 0 \\ v + h\end{pmatrix},$ where

h

$h$ represents the fluctuations of the Higgs field about its average value

v \approx 246 GeV

$v \approx 246 \,\textrm{GeV}$ in the vacuum state (the vacuum expectation value). The quanta of the field

h

$h$ are identified with the Higgs bosons discovered at CERN in 2012.

In the following, we shall also need the conjugate field defined by

H_{c} \equiv i τ_{2} H = \frac{1}{\sqrt{2}} (\begin{matrix} v + h \\ 0 \end{matrix}) .

$H_c \equiv i \tau_2 H = \frac{1}{\sqrt{2}}\begin{pmatrix} v + h \\ 0 \end{pmatrix}.$

v = sm.symbols('v')
h = sm.symbols('h')
H = Matrix([0, (v+h)/sm.sqrt(2)])
F = (g1/2)*YphiL*b + (g2/2)*w
Hc=  sm.I*tau2*H
F, H, Hc

Note¶

The matrix $F$ is Hermitian, that is, $F^\dagger = F$ .

y    = F*H
ybar = H.T*F
f = ybar*y
f = f[0]
f = f.subs(YphiL, 1)
f

Vector boson - Higgs boson interactions¶

f = f.subs(g1*B/2 - g2*W3/2, Z * sm.sqrt(g1**2 + g2**2)/2)
f

The terms involving the Higgs boson field are the single and di-Higgs interactions, while the term $\propto v^2$ is the mass term. For a massive vector boson, $V_\mu$ , the mass term has the form $(m_V^2 \, / \, 2) V_\mu V^\mu$ . The term $W^+_\mu W^{- \mu}$ is the sum of two terms quadratic in the fields $W^1$ and $W^2$ , with the same mass, so the mass term has the form $m_W^2 \, W^+_\mu W^{- \mu}$ . Therefore, in the GSW model the $W$ boson the mass is predicted to be $m_W = v g_2 \, / \, 2$ , while for the $Z$ boson the prediction is $m_Z = v \sqrt{g_1^2+g_2^2} \, / \, 2$ and, therefore, $m_W / m_Z = g_2 / \sqrt{g_1^2 + g_2^2} = \cos\theta_W$ , which modulo higher order corrections agrees with measurements as shown below.

sw2 = 0.2397
sw = sm.sqrt(sw2)
cw = sm.sqrt(1-sw2)
print('cos(theta_W) = %8.3f' % cw)

cos(theta_W) =    0.872

MW=80.385
MZ=91.188
cwp = MW/MZ
print('M_W / M_Z    = %8.3f\t%8.3f' % (cwp, cw / cwp))

M_W / M_Z    =    0.882	   0.989

The Higgs potential¶

$V(H) = \mu^2 H^\dagger H - \lambda (H^\dagger H)^2$

mu, lm = sm.symbols('\mu, \lambda')
V = mu*mu*H.T*H - lm*(H.T*H)**2
V = V.expand()[0]
V

$h$ - $h$ interactions at low energies¶

V.subs(mu**2, v**2*lm)

Yukawa interactions between fermions and the Higgs field¶

The interactions between the fermions and the Higgs field are presumed to be Yukawa interactions,

\begin{aligned} - \sqrt{2} g_{e} (\bar{L} H e_{R} + {\bar{e}}_{R} H^{†} L) - \sqrt{2} g_{u} (\bar{Q} H_{c} u_{R} + {\bar{u}}_{R} H_{c}^{†} Q) - \sqrt{2} g_{d} (\bar{Q} H d_{R} + {\bar{d}}_{R} H^{†} Q) \end{aligned}

$\begin{align*} -\sqrt{2} g_e (\bar{L} H e_R + \bar{e}_R H^\dagger L) -\sqrt{2} g_u (\bar{Q} H_c u_R + \bar{u}_R H_c^\dagger Q) -\sqrt{2} g_d (\bar{Q} H d_R + \bar{d}_R H^\dagger Q) \end{align*}$

Let us pause for a minute to consider the dimensions of the above interactions. In natural units, the Lagrangian has units of $[M]^4$ . Since a fermion mass term looks like

m \bar{ψ} ψ

$m \bar{\psi} \psi$

and $\textrm{dim}(m) = [M]$ , it follows that a fermion field has dimensions $[M]^{3/2}$ . Since the Higgs boson is a scalar field, it has dimensions $[M]$ . Consequently, the Yukawa interactions between fermions and the Higgs boson must be a gauge invariant combination of two fermion fields and the Higgs field. Also, because the Higgs field is a doublet, it is necessary to have it interact with the Dirac conjugate of a doublet field and a singlet field.

Notice also that in order to avoid a flavor changing Yukawa interaction, we need to fudge the interaction of the up quark field with the Higgs field by using the conjugate field $H_c$ rather than $H$ !

ge, me = sm.symbols('g_e, m_e')
gu, mu, gd, md = sm.symbols('g_u, m_u, g_d, m_d')
gu, mu, gd, md
ge, me, gu, mu, gd, md

Hc = sm.I*tau2*H
Qbar, Q, Hc, Hc.T

Le = -ge*(Lbar*H*eR + ebarR*H.T*L)[0]*sm.sqrt(2)
Le = Le.expand()
Le = Le.simplify().expand().collect(h*ge).collect(ge*v)
Le = Le.subs(ge*v, me)
Le = Le.subs(ge, me/v)
Le

Lq = -gd*(Qbar*H*dR  + dbarR*H.T*Q)[0]*sm.sqrt(2) \
     -gu*(Qbar*Hc*uR + ubarR*Hc.T*Q)[0]*sm.sqrt(2)
Lq = Lq.expand()
Lq = Lq.subs(gu*v, mu)
Lq = Lq.subs(gd*v, md)
Lq = Lq.collect(gd*h)
Lq = Lq.collect(gu*h)
Lq = Lq.collect(md)
Lq = Lq.collect(mu)
Lq = Lq.subs(gu, mu/v)
Lq = Lq.subs(gd, md/v)
Lq

We see that the Yukawa couplings, together with the particular choice of the value of the Higgs field in the vacuum state, yields two important consequences as alluded to above: mass terms for the fermions and interactions between the Higgs boson and fermions that are proportional to the fermion mass and inversely proportional to the vacuum expectation value $v$ . We conclude that the Higgs mechanism generates mass terms for the weak vector bosons as well as the fermions while preserving guage invariance.

The Cabibbo Hypothesis¶

So far we have considered only the 1st generation of particles and we have implicitly assumed that there is no "crosstalk" between the generations. However, there are decays that can be explained if "crosstalk" exists between generations. For example, the existence of decays such as

Λ (d, u, s) \to p (d, u, u) e^{-} {\bar{ν}}_{e}

$\Lambda (d, u, s) \rightarrow p (d, u, u) \, \, e^- \, \, \bar{\nu}_e$ are readily explained by supposing that the

W

$W$ boson can convert s-quarks to u-quarks. Further clues are provided by the rates of decays such as

\begin{aligned} π^{+} & \to μ^{+} ν_{μ}, \\ K^{+} & \to μ^{+} ν_{μ}, \end{aligned}

$\begin{align} \pi^+ & \rightarrow \mu^+ \, \nu_\mu,\\ K^+ & \rightarrow \mu^+ \, \nu_\mu, \end{align}$ which are measured to be in the ratio

\begin{aligned} \frac{Γ (π^{+} \to μ^{+} ν_{μ})}{Γ (K^{+} \to μ^{+} ν_{μ})} & \approx \frac{1}{20} . \end{aligned}

$\begin{align} \frac{\Gamma(\pi^+ \rightarrow \mu^+ \, \nu_\mu)} {\Gamma(K^+ \rightarrow \mu^+ \, \nu_\mu)} & \approx \frac{1}{20}. \end{align}$ The above decays can be described with the following Feynman diagrams, decays

in which the couplings

g_{2} ?

$g_2\textrm{?}$ differ from

g_{2}

$g_2$ by the appropriate factor in order to match the measured ratio. These observations and many others were explained by the Italian physicist Cabibbo who suggested that the lower component of the quark doublet, that is, the down quark field, be replaced with the combination

d_{L} \to d_{L}^{'} = \cos θ_{C} d_{L} + \sin θ_{C} s_{L},

$d_L \rightarrow d_L^\prime = \cos\theta_C \, d_L + \sin\theta_C \, s_L,$ where

θ_{C}

$\theta_C$ is called the Cabibbo angle. With a suitable choice of the Cabibbo angle (about 12

^{o}

$^\textrm{o}$ ), the Cabibbo hypothesis worked well. However, as is evident from the form of the down component of the left-handed part of weak neutral current,

\begin{aligned} (I_{3}^{d} - Q^{d} \sin^{2} (θ_{W})) {\bar{d}}_{L} γ^{μ} d_{L}, \end{aligned}

$\begin{align} \left( I^{d}_{3} - Q^{d} \sin^{2}{\left (\theta_{W} \right )}\right) \overline{d}_L \gamma^{\mu} d_{L} , \end{align}$ the Cabibbo hypothesis predicts the existence of flavor-changing neutral currents (FCNC) at a rate in sharp disagreement with observations.

thetaC = sm.symbols('theta_C')
xbarL, xL = sm.symbols('\overline{d^\prime}_L, '\
                       'd^\prime_L', commutative=False)
Nc = xbarL * gamma * xL 
Nc = Nc.subs({xL: sm.cos(thetaC)*dL + sm.sin(thetaC)*sL,
              xbarL: sm.cos(thetaC)*dbarL + sm.sin(thetaC)*sbarL})
Nc = Nc.expand()
Nc = (Id3 - Qd * sm.sin(thetaW)**2) * Nc
Nc

The GIM Mechanism¶

In 1970, Sheldon Glashow, John Iliopoulos and Luciano Maiani (GIM) noted that if the strange quark were the lower component of another doublet,

(\begin{matrix} c_{L} \\ s_{L} \end{matrix}),

$\begin{pmatrix} c_L \\ s_L \end{pmatrix},$ whose upper component would be a new quark field, dubbed "charm", and if one generalized Cabibbo's idea and assumed that the lower components of the two quark doublets were mixed as follows

(\begin{matrix} d_{L}^{'} \\ s_{L}^{'} \end{matrix}) = (\begin{matrix} \cos θ_{C} & \sin θ_{C} \\ - \sin θ_{C} & \cos θ_{C} \end{matrix}) (\begin{matrix} d_{L} \\ s_{L} \end{matrix}),

$\begin{pmatrix} d_L^\prime \\ s_L^\prime \end{pmatrix} = \begin{pmatrix} \cos\theta_C & \sin\theta_C \\ -\sin\theta_C & \cos\theta_C \end{pmatrix} \begin{pmatrix} d_L \\ s_L \end{pmatrix},$ then the replacement

s_{L} \to s_{L}^{'} = - \sin θ_{C} d_{L} + \cos θ_{C} s_{L}

$s_L \rightarrow s_L^\prime = -\sin\theta_C \, d_L + \cos\theta_C \, s_L$ would yield exact cancellation of the unwanted FCNC interactions, as demonstrated below.

Implement GIM Mechanism¶

ybarL, yL = sm.symbols('\overline{s^\prime}_L, '\
                                  's^\prime_L', 
                                  commutative=False)
# add the GIM terms to the Cabbibo terms
yy = ybarL * gamma * yL
yy = yy.subs({yL:-sm.sin(thetaC)*dL + sm.cos(thetaC)*sL,
              ybarL:-sm.sin(thetaC)*dbarL + sm.cos(thetaC)*sbarL})
yy = yy.expand()
Nc += (Id3 - Qd * sm.sin(thetaW)**2) * yy

# and simplify
Nc = Nc.simplify()
Nc

By introducing a 2nd doublet, the GIM mechanism neatly renders the weak neutral currents diagonal again and, moreover, predicts the existence of a 4th quark, the charm quark. This prediction was spectacularly confirmed in 1974 with the discovery of the $J/\psi$ at Brookhaven and SLAC.

Unfortunately, however, no one understands why mixing across the (three) generations should exist and what determines the strength of the mixing.

Vector boson self-interactions¶

The pure vector boson Lagrangian is given by

\begin{aligned} L_{V B} & = - \frac{1}{4} W_{μ ν} \cdot W^{μ ν} - \frac{1}{4} B_{μ ν} B^{μ ν}, \end{aligned}

$\begin{align*} {\cal L}_{VB} & = - \frac{1}{4} \mathbf{W}_{\mu\nu} \cdot \mathbf{W}^{\mu\nu} - \frac{1}{4} B_{\mu\nu} B^{\mu\nu}, \end{align*}$ where

\begin{aligned} (W_{μ ν})_{i} & = (\partial_{μ} W_{ν} - \partial_{ν} W_{μ})_{i} - g_{2} ϵ_{i j k} W_{μ}^{j} W_{ν}^{k}, \\ B_{μ ν} & = \partial_{μ} B_{ν} - \partial_{ν} B_{μ} . \end{aligned}

$\begin{align*} (\mathbf{W}_{\mu\nu})_i & = (\partial_\mu \mathbf{W}_\nu- \partial_\nu \mathbf{W}_\mu)_i - g_2 \, \epsilon_{ijk} W^j_\mu W^k_\nu, \\ B_{\mu\nu} & = \partial_\mu B_\nu - \partial_\nu B_\mu. \end{align*}$

From

\begin{aligned} (\begin{matrix} B_{μ} \\ W_{μ}^{3} \end{matrix}) & = (\begin{matrix} \cos θ_{W} & - \sin θ_{W} \\ \sin θ_{W} & \cos θ_{W} \end{matrix}) (\begin{matrix} A_{μ} \\ Z_{μ} \end{matrix}) \\ W_{μ}^{\pm} & = \frac{1}{\sqrt{2}} (W_{μ}^{1} \mp W_{μ}^{2}) \end{aligned}

$\begin{align} \begin{pmatrix} B_\mu \\ W^3_\mu \end{pmatrix} & = \begin{pmatrix} \cos\theta_W & -\sin\theta_W \\ \sin\theta_W & \cos\theta_W \end{pmatrix} \begin{pmatrix} A_\mu \\ Z_\mu \end{pmatrix} \\ W^\pm_\mu & = \frac{1}{\sqrt{2}} (W^1_\mu \mp W^2_\mu) \end{align}$ we obtain

\begin{aligned} W^{1} & = (W^{-} + W^{+}) / \sqrt{2} \\ W^{2} & = (W^{-} - W^{+}) / \sqrt{(} 2) \\ W^{3} & = \sin θ_{W} A + \cos θ_{W} Z \\ B & = \cos θ_{W} A - \sin θ_{W} Z, \end{aligned}

$\begin{align*} W^1 & = (W^- + W^+)/\sqrt{2}\\ W^2 & = (W^- - W^+)/\sqrt(2)\\ W^3 & = \sin\theta_W A + \cos\theta_W Z\\ B & = \cos\theta_W A - \sin\theta_W Z, \end{align*}$ which we use below to rewrite the vector boson Lagrangian in terms of the usual fields.

Bmu, W1mu, W2mu, W3mu = sm.symbols("B_\mu, W^1_\mu, W^2_\mu, W^3_\mu",
                                   commutative=False)
Bnu, W1nu, W2nu, W3nu = sm.symbols("B_\\nu, W^1_\\nu, W^2_\\nu, W^3_\\nu",
                                   commutative=False)
Wplusmu, Wminusmu     = sm.symbols("W^+_\mu, W^-_\mu",
                                   commutative=False)
Wplusnu, Wminusnu     = sm.symbols("W^+_\\nu, W^-_\\nu",
                                   commutative=False)
dmu, dnu              = sm.symbols('\partial_\mu, \partial_\\nu', 
                                   commutative=False)
Zmu, Amu              = sm.symbols('Z_\mu, A_\mu',
                                   commutative=False)
Znu, Anu              = sm.symbols('Z_\\nu, A_\\nu',
                                   commutative=False)
dmuW1nu = dmu*W1nu
dmuW2nu = dmu*W2nu
dmuW3nu = dmu*W3nu

dnuW1mu = dnu*W1mu
dnuW2mu = dnu*W2mu
dnuW3mu = dnu*W3mu

dmuBnu  = dmu*Bnu
dnuBmu  = dnu*Bmu

Vmu = Matrix([W1mu, W2mu, W3mu])
Vnu = Matrix([W1nu, W2nu, W3nu])

dVmu= Matrix([dmuW1nu, dmuW2nu, dmuW3nu])
dVnu= Matrix([dnuW1mu, dnuW2mu, dnuW3mu])
WW  = dVmu - dVnu - g2*Vmu.cross(Vnu)
WW  = WW.subs({W1mu: (Wminusmu + Wplusmu)/sm.sqrt(2), 
               W1nu: (Wminusnu + Wplusnu)/sm.sqrt(2),
               W2mu: (Wminusmu - Wplusmu)/sm.sqrt(2), 
               W2nu: (Wminusnu - Wplusnu)/sm.sqrt(2),
               W3mu: sm.sin(thetaW)*Amu + sm.cos(thetaW)*Zmu,
               W3nu: sm.sin(thetaW)*Anu + sm.cos(thetaW)*Znu}).expand()

BB  = dmuBnu - dnuBmu
BB  = BB.subs({Bmu: sm.cos(thetaW)*Amu - sm.sin(thetaW)*Zmu,
               Bnu: sm.cos(thetaW)*Anu - sm.sin(thetaW)*Znu}).expand()
VV  = BB*BB +  (WW.T*WW)[0]
VV  = VV.expand()
VV = trigsimp(VV.subs({g2: q / sm.sin(thetaW)}))
VV

This is a complicated expression, but some conclusions are readily apparent such as the prediction of 3 and 4 vector boson interactions.

The QCD Lagrangian¶

The Lagrangian of quantum chromodynamics (QCD), for the 1st generation of particles, is given by

\begin{aligned} L_{QCD} & = \bar{U} i γ^{μ} (\partial_{μ} 1 + i g_{3} \frac{λ}{2} \cdot G_{μ}) U + \bar{D} i γ^{μ} (\partial_{μ} 1 + i g_{3} \frac{λ}{2} \cdot G_{μ}) D \\ - \frac{1}{4} G_{μ ν} \cdot G^{μ ν} \\ and \\ (G_{μ ν})_{a} & = (\partial_{μ} G_{ν})_{a} - (\partial_{ν} G_{μ})_{a} - g_{3} f_{a b c} G_{μ}^{b} G_{ν}^{c}, \end{aligned}

$\begin{align*} {\cal L}_\textrm{QCD} & = \overline{U} i \gamma^\mu \left(\partial_\mu \mathbf{1} + i g_3 \frac{\boldsymbol{\lambda}}{2} \cdot \mathbf{G}_\mu\right) U + \overline{D} i \gamma^\mu \left(\partial_\mu \mathbf{1} + i g_3 \frac{\boldsymbol{\lambda}}{2} \cdot \mathbf{G}_\mu\right) D \\ & - \frac{1}{4} \mathbf{G}_{\mu\nu} \cdot \mathbf{G}^{\mu\nu}\\ \textrm{and } & \\ (\mathbf{G}_{\mu\nu})_a & = (\partial_\mu \mathbf{G}_\nu)_a - (\partial_\nu \mathbf{G}_\mu)_a - g_3 \, f_{abc} G^b_\mu G^c_\nu, \end{align*}$

where $U$ and $D$ , respectively, are the up and down quark field triplets each of whose components is associated with one of three quark colors, $\lambda^a$ are the 8 Gell-Mann matrices each associated with one of the 8 gluon fields $G^a_\mu$ , and $g_3$ is the strong coupling parameter. The quantities in parentheses are $3 \times 3$ matrices. The QCD Lagrangian is invariant under $SU(3)$ transformations and the numbers $f_{abc}$ are the structure constants of the group.

When QCD is added to the GSW (electroweak) Lagrangian, thereby forming the Lagrangian of the Standard Model (SM), the up and down quark fields acquire a 3-color index. Therefore, in the electroweak part of the Lagrangian each quark field will now carry a color index, in which case it is necessary to sum over the color indices in that part of the Lagrangian. Note that each quark field in the QCD Lagrangian is the sum of left-handed and right-handed fields.

Since the SM is gauge invariant, it is necessary to fix the definition of its bosonic (gauge) fields by choosing a gauge. Choosing a gauge is akin to choosing a coordinate system in that the predictions of the theory do not depend on the choice of gauge, at least in the limit of exact calculations. (In practice, calculations are done using perturbation theory. Consequently, depending on how the calculations are done there may be some residual, unphysical, gauge dependence in the predictions.) Unfortunately, choosing the gauge requires the addition of quite a bit of extra structure, involving unphysical fields called ghost fields, which makes what is otherwise an elegant structure much less so.